======Crystallographic point groups, free electron model======


Starting from 2006 Hafnium silicates replaced SiON as gate oxide
in MOSFETS. The high dielectric constant of HfO2 and the ability
of HfO2 to form silicates played a key role in the industrial transition.

<note importat>
please download
 **hfo2.py, PhysRevB...pdf, bands_si.py, bands_cu.py** from [[https://polybox.ethz.ch/index.php/s/CH5VdcI40YdELez|this link]] 
</note>

====Task1====

Construct the primitive cell of the monoclinic phase of HfO2, using ASE and teh information contained in the HfO2 manuscript

<note tip>
Follow the instructions contained in the ASE webpage dedicated to spacegroups

[[https://wiki.fysik.dtu.dk/ase/ase/spacegroup/spacegroup.html]]

and find in this article (a pdf copy is included in dropbox)

[[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.65.233106]]

all data necessary to construct the monoclinic phase of HfO2


modify the python script hfo2.py according to your needs.

(replace "...")

**(comment the line "hfo2.write("hfo2.png")")**

Execution of the python script 
<code>
python hfo2.py
</code>

will open the ASE visualizer showing you the structure,
will produce the file hfo2.xyz, (hfo2.png) 
</note>

<note important>
-how many atoms are contained in the unit cell?

-compute the volume of the unit cell
</note>

====Task2====
Have a look at the atomic coordinates, for example in the file hfo2.xyz (where you also find the cell vectors in cartesian coordinates) and 
<note important>
try to reproduce them (just the 4 Hf atoms), starting from the
coordinates that you find in the article and applying the symmetry operations of the space group:

1)x,y,z

2)-x,y+1/2,-z+1/2

3)-x,-y,-z

4)x,-y+1/2,z+1/2
<note tip>
**DO NOT FORGET Periodic Boundary Conditions!!** but it's simple: x,y,z are given in crystal coordinates so, if for example x=0.276
for "-x" you can use -x+1 = 0.724


usually the coordinates are provided in crystal coordinates so
if a1=(a1x,a1y,a1z),a2=(a2x,a2y,a2z),a3=(a3x,a3y,a3z) are the three basis vectors of the crystal in cartesian coordinates
and (x1,y1,z1)  the crystal coordinates of atom 1, the cartesian coordinates of atom 1 will be:
x1*a1 + x2*a2 + x3*a3
</note>
</note>
<note important>obtain teh HfO2 structure from [[https://materialsproject.org/]]</note>

====Task 3====
Check the lecture notes for the free electron model and:
<note important>Compute the Fermi energy (in eV) and the Fermi wavevector (in cm-1) for Cu,Au,Ag </note>


====Task 4====
Have  a look at this ASE page to compute bandstructures and the symmetry points of the Brillouin zone of a crystal:

[[https://wiki.fysik.dtu.dk/ase/ase/dft/kpoints.html]]

<note important>
Compute the free electron bandstructure of Si and Cu
Write the CARTESIAN COORDINATES of the Gamma, X, W points of FCC
<note tip>
Have a look at the scripts included in the exercise directory

</note>
<note warning>How does the Fermi energy that you computed for Cu compares with the one plotted in the bands?</note>
</note>